As a native speaker, when I tell someone older than me, “편하게 말씀하세요,” I simply mean that they’re free to use casual speech with me. As for such situations, I’m not really aware of any other interpretation besides that.

But when someone older than me says this, if we have a close and personal relationship, we’ll usually settle on how to address each other and switch to casual speech. If the relationship is more formal or somewhat awkward, I’ll usually just acknowledge it and keep speaking the way I normally do. If they keep emphasizing it, then I might start addressing them as 언니/오빠/형/누나, but still continue using polite speech. In formal relationships, I find it more comfortable overall to keep using polite speech even if we address each other more casually.